NAKAYAMA CATEGORIES AND GROUPOID QUANTIZATION FABIO TROVA Abstract. We provide a precise description, albeit in the situation of standard categories, of the quantization functor Sum proposed by D.S. Freed, M.J. Hopkins, J. Lurie, and C. Teleman in a way enough abstract and flexible to suggest that an extension of the construction to the general context of (1; n)-categories should indeed be possible. Our method is in fact based primarily on dualizability and adjunction data, and is well suited for the homotopical setting. The construction also sheds light on the need of certain rescaling automorphisms, and in particular on the nature and properties of the Nakayama isomorphism. Contents 1. Introduction1 1.1. Conventions4 2. Preliminaries4 2.1. Duals4 2.2. Mates and Beck-Chevalley condition5 3. A comparison map between adjoints7 3.1. Projection formulas8 3.2. Pre-Nakayama map 10 3.3. Tensoring Kan extensions and pre-Nakayama maps 13 4. Nakayama Categories 20 4.1. Weights and the Nakayama map 22 4.2. Nakayama Categories 25 5. Quantization 26 5.1. A monoidal category of local systems 26 5.2. The quantization functors 28 5.3. Siamese twins 30 6. Linear representations of groupoids 31 References 33 arXiv:1602.01019v1 [math.CT] 2 Feb 2016 1. Introduction A k-extended n-dimensional C-valued topological quantum field theory n−k (TQFT for short) is a symmetric monoidal functor Z : Bordn !C, from the (1; k)-category of n-cobordisms extended down to codimension k [CS, Lu1] to a symmetric monoidal (1; k)-category C. For k = 1 one recovers the classical TQFT's as introduced in [At], while k = n gives rise to fully Date: October 22, 2018. 1 2 FABIO TROVA extended TQFT's which, by the Cobordism Hypothesis [BD1, Lu1], we know being in correspondence with the fully dualizable objects of C. In [FHLT] the authors suggested the existence of an extended TQFT of Dijkgraaf-Witten type, associated to any given representation V : X !C of a finite 1-groupoid X with values in a symmetric monoidal (1; k)-category C; under the identification of 1-groupoids with homotopy types, V is equiv- alently seen as the datum of a local system on a topological space X of finite homotopy type, with values in C. Such a theory appears as the composition of two monoidal functors n−k Bunk;V : Bordn ! Famk(C) Sumk : Famk(C) !C where Famk(C) is an (1; k)-category of representations, with objects spaces equipped with a functor to C and morphisms the correspondences between these (see [Ha2, Lu1]). In the fully extended case the first functor associates to the point the given representation V , while Sum is the proper quantization functor carrying a representation W : Y !C to its colimit. This can be thought of as taking the global sections of the local system W , or, from a more algebraic point of view, as the space of coinvariants for the action of the 1-groupoid Y on W . Unfortunately in [FHLT] not many details are provided on how the above machinery should behave; the article in fact only addresses the above colimit construction and demands that colimits and limits in C coincide in order to get a well defined functor. A better picture is given in the case when C is the (1; n)-category Algn−1 (in the sense of [FHLT, Def. 7.1]) of iterated algebras and modules over Vect, and V : X ! Algn−1 is the trivial rep- resentation collapsing everything on the unit. In this situation, a detailed definition of the functor Sum has been given for n = 2 in [Mo1], while in [Mo2] also spaces endowed with a 2-cocycle have been considered; again, the author requires the existence of an isomorphism between right and left adjoints, given explicitly by the Nakayama isomorphism (cfr. [Ben]). It is worth noting that, despite the Cobordism Hypothesis, neither [FHLT] or [Mo1, Mo2] seem to make use of dualizability in their construction. The primary goal of this paper is to provide a detailed construction of the functor Sum1 and therefore of n-dimensional theories in the sense of Atiyah. The method we propose differs from the ones above in two main points: - It does not assume the existence of any a priori given isomorphism between right and left adjoints. - It relies almost only on existence of duals, in line with the Cobordism Hypothesis. The main obstacles to the definition of the functor Sum appear when studying its behaviour at the level of morphisms. In fact, to a morphism α in Famn(C) we could associate a morphism Sum(α) in C via a pull-push pro- cedure, as soon as we have a natural transformation (which is not required to be invertible) f∗ ! f! from the right to the left adjoint of the pullback functor f ∗, for any map f : X ! Y of spaces. NAKAYAMA CATEGORIES AND GROUPOID QUANTIZATION 3 A candidate γf is presented in Definition 3.6; the map was originally de- fined in [Ha1] as long as certain projection formulas were satisfied, which actually always happens (Propositions 3.1 and 3.2) in presence of duals. De- spite being very well behaved with respect to homotopy squares and external tensor products (as shown in sections 3.2 and 3.3), the map γ still lacks an essential feature. In fact, in order to make Sum into a functor, we need the assignment f 7! γf to be functorial as well, in the sense that for any two f g composable maps X −! Y −! Z the following triangle g!f∗ γg ? ? g!γf ?? ? g∗f∗ / g!f! γgf should commute. It can be seen that such condition is generally not verified, due to the possible non-triviality of the fibers of the maps f; g involved. As shown in Examples 4.4 and 4.5, when the target category C is that of finite vector spaces over a positive characteristic field, one of the two paths might be the zero map while the other one is nonzero. Instead, in characteristic zero, the two ways of going from the right to the left adjoints differ by an invertible scalar. One might then hope the assignment f 7! γf to define a \projective functor", and the discrepancy to result in a \cocycle condition". This is actually the situation in the case of characteristic zero fields: we can in fact −1 rescale the above map γ by a factor δf related to the Baez-Dolan groupoid cardinality [BD2], so to neutralize the anomaly. In categorical terms, δf can be described via γ itself (Definition 4.7) and we can define a new map (the Nakayama map, Definition 4.8) νf : f∗ ! f! −1 by multiplying γf by δf . The best one can now achieve is to ask this new map to be functorial in the above sense. We will therefore define a Nakayama category (Definition 4.13) as a rigid category, in which for any map f of spaces the endomorphism δf is invertible and the assignment f 7! νf is functorial. Under these hypotheses in Theorems 5.1 and 5.2 we will construct, by means of Kan extensions, two monoidal quantization functors Sum : Fam1(C) !C Prod : Fam1(C) !C The map ν will then define a monoidal transformation between them and turn out to be an isomorphism (Proposition 5.3 and Corollary 5.5), thus answering an ambidexterity problem [HeL, HoL] in the context of categories with duals. As one would expect, when C is the category of finite dimensional vec- tor spaces over a characteristic zero field, the map νf recovers the known Nakayama isomorphism, and justifies the choices made in [FHLT, Mo1, Mo2] as instances of a more general and abstract framework. In particular, it ex- plains from a new point of view the obstruction to quantizing over fields of 4 FABIO TROVA positive characteristic. Although the present paper only deals with standard categories and group- oids, the methods and ideas we propose are intrinsically homotopical, and formulated mainly in terms of adjunction and duality data via calculus of mates and Beck-Chevalley conditions. This suggests it should be possible to export them to the framework of higher categories so to build the proper n-dimensional theories which are the true aim of [FHLT], a topic which will be investigated elsewhere. 1.1. Conventions. While most of the constructions can be stated in the general case of non-symmetric monoidal categories, we will be interested into symmetric ones. By monoidal category we will therefore mean symmetric monoidal category. By space we will always mean 1-groupoid; in particular, also groupoids (i.e. homotopy 1-types) will be often referred to as spaces. Aknowledgements. I am happily in debt with Domenico Fiorenza, who supervised my PhD thesis from which part of this work is extracted; I'm glad to thank him for the many and helpful discussions, during and after my PhD. I would like to thank as well Claudia Scheimbauer, Urs Schreiber and Christoph Schweigert for interesting and fruitful conversations and com- ments. 2. Preliminaries We quickly review here below some notions and results, which are essential to the development of our machinery. 2.1. Duals. Recall first of all the notion of dualizable object in a monoidal category C, which generalises and axiomatises the familiar concept of dual of a vector space. Dualizability is a key tool in this work, and more generally the main and possibly only ingredient to the construction of an extended Topological Quantum Field Theory, which is the ultimate goal of this paper.